Large deviations for symmetric Markov processes and Dirichlet forms

对称马尔可夫过程和狄利克雷形式的大偏差

• 批准号: 15540103
• 负责人: TAKEDA Masayoshi
• 金额: $ 2.3万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540103/
• 关键词: Markov process; Dirichlet form; Feynman-Kac functional; time-changed process; ファインマンーカッツ汎関数; ファインマン-カッツの公式; 対称マルコフ過程; 大偏差原理; 分枝対称マルコフ過程; 対称安定過程

We proved that the integrability of Feynman-Kac functionals (gaugeability) is equivalent to that the principal eigenvalue of time-changed process is greater than 1, which is also equivalent to the subcriticality of Schroedinger operators. This fact says that the principal eigenvalue of time-changed process accurately measures the size of measures. Using this fact, we obtained three results : The first result is that the ultracontractiyity of Schroedinger semigroups holds if and only if the princilal eigenvalue of time-changed process is greater than 1. The second result is that the expectation of the number of branches hitting a closed set in a branching symmetric stable process is finite if and only if the princilal eigenvalue is greater than 1. The final result is as foolows : Suppose that the heat kernel on a complete Riemannian manifold satisfies the global Gaussian bounds, so called Li-Yau estimate. Then the heat kernel of the Schroedinger operator also possesses the global Gaussian bounds, if and anly if the princilal eigenvalue is greater than 1.

我们证明了Feynman-Kac泛函的可积性（可测性）等价于时变过程的主特征值大于1，也等价于薛定谔算子的亚临界性。这个事实表明，时变过程的主特征值准确地测量了测度的大小。利用这个事实，我们得到了三个结果：第一个结果是薛定谔半群的超收缩性当且仅当时变过程的主特征值大于 1 时成立。第二个结果是分支命中数的期望当且仅当主特征值大于 1 时，分支对称稳定过程中的闭集是有限的。最终结果如下：假设热核完整的黎曼流形满足全局高斯边界，即所谓的 Li-Yau 估计。那么当且仅当主特征值大于 1 时，薛定谔算子的热核也具有全局高斯界。

项目成果

Subcriticality and gaugeability for symmetric α-stable processes

对称 α 稳定过程的亚临界性和可测量性

• 发表时间: 2004
• 期刊: Forum math. 16
• 作者: 竹田雅好;服部哲弥;塩谷隆;M.Takeda;T.Hattori;T.Shioya;服部哲弥;塩谷隆;竹田 雅好;竹田雅好
• 通讯作者: 竹田雅好

Absolute continuity of symmetric Markov processes

对称马尔可夫过程的绝对连续性

• 发表时间: 2004
• 期刊: Ann.Probab. 32
• 作者: 竹田雅好;服部哲弥;塩谷隆;M.Takeda;T.Hattori;T.Shioya;服部哲弥;塩谷隆;竹田 雅好;竹田雅好;M.Takeda;竹田 雅好
• 通讯作者: 竹田 雅好

竹田 雅好: "Criticality of generalized Scbvodinger operators and differentiability of spectral functious"Advanced Studies in Pure Mathematics. 41. 333-350 (2004)

Masayoshi Takeda：“广义 Scbvodinger 算子的临界性和谱函数的可微性”纯数学高级研究 41. 333-350 (2004)。

竹田 雅好: "Subcriticality and Gaugeability for symmetric α-stable processes"Fourm Math.. (to appear).

Masayoshi Takeda：“对称 α 稳定过程的亚临界性和可测性”Fourm Math..（即将出现）。

Variational formula for Dirichlet forms and estimates of principal eigenvalues for symmetric α-stable process

对称α稳定过程的狄利克雷形式的变分公式和主特征值的估计

• 发表时间: 2005
• 期刊: Potential Analysis (in press)(to appear)
• 作者: 竹田雅好;服部哲弥;塩谷隆;M.Takeda;T.Hattori;T.Shioya;服部哲弥;塩谷隆;竹田 雅好
• 通讯作者: 竹田 雅好